The curvature 1/R at any point of the curve representing the deformed shape of a beam is given by:
Try it first: pick an option, then read the explanation.
Correct answer: B. ±(d²y/dx²) / [1 + (dy/dx)²]^(3/2)
Asked in SSC JE 2013
Detailed Explanation
This is the exact curvature formula from calculus for any plane curve y(x). Beam theory then uses the small-slope approximation.
\[ \frac{1}{R} = \pm\frac{\dfrac{d^2y}{dx^2}}{\left[1 + \left(\dfrac{dy}{dx}\right)^2\right]^{3/2}} \approx \frac{d^2y}{dx^2}\quad(\text{small slopes}) \Rightarrow \frac{M}{EI} = \frac{d^2y}{dx^2} \]
Why “±(d²y/dx²) / [1 + (dy/dx)²]^(3/2)” Is Correct
Option (b) is the exact curvature. Because beam slopes are tiny (e.g. 0.002 rad), (dy/dx)² is negligible and 1/R ≈ d²y/dx², which gives EI y'' = M.
Why the Other Options Are Wrong
- A. ±(dy/dx) / [1 + d²y/dx²]^(1/2): Uses the slope in the numerator and the wrong power.
- C. ±(d²y/dx²) / [1 + d²y/dx²]^(1/2): The denominator must contain the square of the slope, not the second derivative.
- D. ±(dy/dx) / [1 + d²y/dx²]²: Wrong numerator and wrong power.
Key Point
1/R = y''/(1 + y'²)^(3/2) ≈ y'' for small deflections.
Key Concepts for Students
- The small-slope assumption is what makes beam deflection theory linear.
- Bending equation: M/I = E/R.
- Large-deflection problems (elastica) need the exact formula.
Practice more: this is question 37 in Slope and Deflection MCQs – Part 3. Try the whole set and check every answer instantly.